Questions tagged [billiards]

For questions about billiards; dynamical systems involving point particles (billiard balls) that travel in straight lines on the interior of some bounded domain (the billiard table) and experience perfectly elastic reflections on collision with boundary.

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Can billiards fill any figure?

For any 2D closed shape, is there always a point and angle for which the billiard movement of a ball fully completes the interior of the shape? In other words, does the path eventually converge to the ...
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Can ellipsoid or other surface trap incoming light or photon

There had already several questions about light-trapping curves, such as drawing a curve captures light or photon-trapping curve, and well known cases had been answered such as ellipse or hyperbola. ...
Mountain's user avatar
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Proof that Polygonal Billiards Have Zero Entropy

I'm reading this paper "Billiards In Polygons" by Boldrighini et al. They say that polygonal billiards have zero measure-theoretic entropy, because a given element of the configuration space ...
interstice's user avatar
-1 votes
1 answer
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Billiard problem variation [duplicate]

Suppose a square table. You are shooting from a corner. A ball ends its path upon landing in a corner. At what angles can you shoot a ball from your corner such that the ball will never come back to ...
redrobinyum's user avatar
5 votes
1 answer
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If $D$ isn't a convex billiard table, then the billiard map $T$ is not continuous

I am currently reading chapter 3 of "Geometry and Billiards" by Serge Tabachnikov, and I have some doubts about the need of using convex sets. So, here's how the billiard map is defined: To ...
ImHackingXD's user avatar
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Construct a mirrored rectangle tunnel unilluminable from one end to another, consisting of only orthogonal rectangles and circles

In a problem there consist of a 2D tunnel, a rectangle with two opposite sides removed as its faces, in which obstacles are only orthogonal rectangles (those whose edges are parallel to one of the ...
Alex-Github-Programmer's user avatar
3 votes
1 answer
137 views

Reference request: Why is an integrable system called an integrable system and why is the dynamical billiard on a disk completely integrable?

I am seeking detailed reference or references to help me understand the following: Relevant history and motivation behind the term "integrable system" with appropriate primers The meaning ...
Cartesian Bear's user avatar
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Elliptical billiards and eccentricity

I have to do an exploration in maths. I was thinking to do it on how the angle of incidence and reflection vary with the eccentricity of the ellipse. Do you think could be interesting? Do you have any ...
Gabriele Carniel's user avatar
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Secure Polygons

The secure-square problem is this: Suppose you have a square with mirror edges. In these square, you choose two points, one as a light source and the other as a target. On these square, you can place ...
Douglas's user avatar
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Alhazen Billiard problem in 3d

I'm sure that everyone knows what the Alhazen's Billiard problem which is basically determining the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
DSONSD's user avatar
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3 answers
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Bound on angle implicitly defined via trigonometric equation

Introduction / setup: In the article Spectral Gap for a Class of Random Billiards by Renato Feres and Hong-Kun Zhang (link: https://www.math.wustl.edu/~feres/spectralgapnew.pdf) a trigonometric ...
Nelus127's user avatar
1 vote
1 answer
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Trigonometric relation between angles in billiard dynamics

Introduction: In the article Spectral Gap for a Class of Random Billiards by Renato Feres and Hong-Kun Zhang (link: https://www.math.wustl.edu/~feres/spectralgapnew.pdf ), proposition 8 (page 20) does ...
Nelus127's user avatar
6 votes
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Why a measure over the points of a billiard which reflect infinitely often in finite time vanish, i.e. why $\int_{N\cap M}f(x^{-})d\mu_1(x^{-1})=0$

The crux of my question is that why $\mu(N^{(2)}) = 0$ when $N^{(2)}$ consists all points of a billiard that reflect infinitely often in finite time under a given flow (i.e. transformation) function. ...
Epsilon Away's user avatar
4 votes
1 answer
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For arbitrary billiard tables with elastic boundary reflections, is the "Lebesgue measure" an invariant of the flow maps?

This question pertains to billiard dynamics and their invariant measures. Specifically, it concerns the oft-quoted 'fact' that billiard flow maps (built using specular reflection boundary conditions) ...
mark_wilkinson's user avatar
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Particular values for the sum of divisors function from billiards

In this post we consider as reference the article Arithmetic billiards from Wikipedia. We consider the arithmetic billiard that is explained in the article. I've wondered if we can compute some simple ...
user759001's user avatar
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Motivation for the definition of a compact Riemann manifold with piecewise smooth boundary

I am asking for either motivation on the requirement regarding $f_i^{-1}(0)$ in the following definition, or better yet a reference to a book dealing with this subject. The following is the definition ...
Epsilon Away's user avatar
3 votes
1 answer
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Using a simulation to show that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit

In 2009, Richard Schwartz proved that any obtuse triangle whose largest angle is $\leq100^{\circ}$ has a stable periodic billiard orbit. My question then, is: How can I reproduce Schwartz's result ...
rb3652's user avatar
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2 votes
1 answer
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Proving a billiards bank shot procedure gives equal angles of incidence and reflection

One method of making a bank shot in billiards involves imagining two lines, the so called "cross-pocket" and "cross-ball" lines: One then projects their point of intersection to ...
Wilder Boyden's user avatar
1 vote
2 answers
137 views

Law of reflection when tangent is undefined

The law of reflection also holds for non-plane mirrors, provided that the normal at any point on the mirror is understood to be the outward pointing normal to the local tangent plane of the mirror at ...
r f's user avatar
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4 votes
1 answer
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Application of billiards

Studying billiards is a difficult problem in general, even in pretty simple cases it has plenty of interesting properties. I would like to understand what can be applications (mathematically or in ...
Wolker's user avatar
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Rectangular billiards: ergodic or not?

I'm approaching dynamical billiards theory, and I've found that rectangular/square billiards are among the examples of integrable and non-ergodic ones. But in this contribution it is (resolutely) said ...
Lo Scrondo's user avatar
3 votes
0 answers
109 views

Sinai billiard isomorphic to hard spheres problem

I have heard that Sinai's famous results on the ergodicity of $n$ hard spheres defined on a $d$ dimensional hypercube with periodic boundaries is isomorphic to a billiard (from which it was proved), ...
algae's user avatar
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6 votes
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Phase space and collision map of a billiard

I am working through Ch. 2 of Chaotic billiards by Chernov and Markarian. A few things are puzzling me about the basic construction and definitions. 2.5. Phase space for the flow. The state of a ...
algae's user avatar
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Number of rays intersecting at a point inside a polygon

I'm working on a project to do with bouncing rays inside polygons and now I've reached a crucial stage of this project in which I need help with and is related to the problem stated below. Your help ...
JCr's user avatar
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1 answer
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Billiards in an ellipse -- a query from a proof

I am reading about billiards in an ellipse and am a little stuck on a couple of points in the following proof (see image). My main doubt is the highlighted line at the bottom: "Notice that line $...
WhySee's user avatar
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2 votes
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Why is the existence of periodic orbits (in triangular billiards) so hard to prove?

I don't get why proving that every triangular billiard has a periodic orbit should be that hard. I mostly understand the partial results on the matter, mainly Every triangle with rational angles ...
augustoperez's user avatar
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8 votes
2 answers
483 views

Bouncing Bullet Problem

This is a problem that was presented to me through the google foobar challenge, but my time has since expired and they had decided that I did not complete the problem. I suspect a possible bug on ...
Kraigolas's user avatar
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5 votes
1 answer
305 views

Do these balls collide?

Assume that two balls $B_1,B_2$ of radius $r$ continuously move around inside of a square of size $d$. They bounce off the walls, i.e. the $x$-component of the velocity is multiplied with $-1$ when ...
Martin Brandenburg's user avatar
5 votes
1 answer
203 views

How to show that the Billiard flow is invariant with respect to the area form $\sin(\alpha)d\alpha\wedge dt$

Consider a plane billiard table $D \subset \mathbb{R}^2$ (i.e. a bounded open connected set) with smooth boundary $\gamma$ being a closed curve. Next, let $M$ denote the space of tangent unit vectors $...
Quoka's user avatar
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3 votes
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Texts on Mathematical Billiards

I want to study the theory of mathematical billiards, and was looking for a text for self-study. If you have a text in mind that is more general i.e. on dynamical systems as a whole but still contains ...
akad137's user avatar
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Verifying that a billiard system satisfies the twist condition.

I'm numerically studying a 2D billiard system whose domain is a unit outer circle with an inner elliptical scatterer of variable geometry. EDIT: Both the circle and ellipse share a common centre. ...
Gregory Ionovich Page's user avatar
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1 answer
99 views

Why are these angles equal?

-see picture above- (from Billiards and Geometry by Serge Tabachnikov) I don't understand why the angles $F_2BA_1$ and $F_1BA_0$ are equal (I do understand the conclusion, that follows from the ...
user avatar
5 votes
1 answer
170 views

Billiards in a holey square

Suppose you start a point-billiard (or light ray) in a square at a random location, shooting off at a random angle, reflecting with angle-of-incidence equals angle-of-reflection. In general, because ...
Joseph O'Rourke's user avatar
1 vote
1 answer
81 views

Chaotic system?

I'm currently working on an individual project. In which I have created a program to simulate a particle bouncing around a finite bordered region. Where I need help in understanding how I might show ...
data_is_fun's user avatar
0 votes
1 answer
36 views

Joining boundary points through affine/billiards trajectories

Assume we have a smooth bounded domain $\Omega \subset \mathbb{R}^d$, and two points at the boundary $x, y$. I want to show that there exists an integer $n$ such that one can draw an affine path ...
Gâteau-Gallois's user avatar
3 votes
1 answer
179 views

Billiard ball mental patients.

The Question: Suppose there are $n$ extremely paranoid, vulnerable mental patients at a hospital. Each day at the lunch hour, they move around like frictionless billiard balls of radius $\rho$ metres ...
Shaun's user avatar
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1 vote
0 answers
186 views

Unfolding a billiard trajectory on an unbounded billiard table

A common "trick" used in the study of polygonal billiards is to unfold the trajectory. I'm interested in billiards which may be bounded in one direction, and unbounded in another. For instance, ...
Joppy's user avatar
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1 vote
1 answer
67 views

Reflecting a vector within a box (variation on the billiards problem)

Let there be a box, with bottom left corner at (0,0), and top right corner being at (m,n), where m and n are positive integers. A starting point is chosen at random within the box at (x,y), such ...
A. Ryan's user avatar
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4 votes
2 answers
1k views

The simplest billiards problem

The Problem: Let's say we have a rectangle of size $m \times n$ centered at the origin (or, if it makes the math easier, you can place it wherever on the plane). We take a billiard ball, ...
NMister's user avatar
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0 answers
125 views

Simple Billiard Table with Positive Lyapunov exponent

I can understand the Lyapunov exponent for a circular billiard table is zero. However, if I were right, I read the Lyapunov exponent for a stadium billiard table is non-zero which was surprising for ...
mathvc_'s user avatar
  • 309
1 vote
1 answer
175 views

Trajectories in circular Billards

Given two points $p_1,p_2$ on a circular billard table. I want to know all billard trajectories from $p_1$ to $p_2$ hitting the boundary precisely once. Model of the circular billard: Denote the ...
Near's user avatar
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3 votes
1 answer
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Natural projection of the billiard phase space

Setup The phase space of the billiard flow $\left\lbrace \Phi^t \right\rbrace$ ($t \in \mathbb{R}$) is given by $\Omega = \mathcal{D} \times S^1$ where $\mathcal{D}$ is a planar billiard table in $\...
Joppy's user avatar
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1 answer
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About a theorem by Mather on smooth convex billiards

I'm a masters student and in about a week I begin working in a short presentation of "a theorem" by John N. Mather. More precisely, I have to give some elements of the proof of a theorem that states ...
Mau_Ssy7's user avatar
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1 answer
207 views

Infinitely many triangular numbers which are of form $n^2-1$

After playing a round of pool billiard recently, I noticed that the fifteen colored balls can be arranged in a triangle (obviously) and all sixteen balls (including the white ball) can be arranged in ...
Simon Bohnen's user avatar
6 votes
2 answers
5k views

Launching billiard balls at 45 degree angles and bouncing of off edges.

Say I have a billiard ball and I launch it from the bottom-left corner of a table with length $x$ and width $y$. Given $x$ and $y$ will the ball reach a corner again, which corner and in how many ...
mtheorylord's user avatar
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2 votes
0 answers
106 views

Number of possible rays in rectangular mirror room between two points

Question There is a rectangular room with mirrors of $a$ width and $b$ height (Aligned with co-ordinate axes, so lower left corner is at $(0, 0)$ and upper right corner is at $(a, b)$). There are two ...
user189237's user avatar
2 votes
0 answers
67 views

Quantitative estimates on space filling curves

To my understanding, quantitative topology/geometry makes statements quantitative. Examples: 1. a quantitative version of Invariance of Dimension is waist inequality. 2. Lusternik-Fet says a closed ...
random_shape's user avatar
2 votes
1 answer
484 views

Tangent to hyperbola meeting on ellipse

Can anyone prove that the two tangent lines to hyperbola that meet on a confocal ellipse make the same angle with the tangent of the ellipse at the point of intersection? Thanks
I.Wms's user avatar
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1 vote
1 answer
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Area of a billiard and its preservation

From S. Tabachnikov's Geometry and billiards Consider a plane billiard table $D$ whose boundary is a smooth closed curve $γ$. Let $M$ be the space of unit tangent vectors $(x, v)$ whose foot ...
Human's user avatar
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2 votes
1 answer
733 views

About trajectories on a circular billiard

Let consider a circular billiard. You start by putting the ball on the edge off the table. The trajectory then is quite simple (see the picture), with equal angles at each rebound. Which will ...
E. Joseph's user avatar
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